12 votes
Accepted

Which algorithms do robo-advisors use?

After having done a lot of research on the topic I found the following excellent research piece on ETF.com: Wealthfront modifies historic asset-class returns with current market implied expected ...
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  • 26.9k
11 votes
Accepted

Calculating alpha and its meaning

Alphas from a time-series regression are error terms in the cross-sectional, linear relationship between expected returns and factor betas. If a factor model were correct those error terms (the alphas)...
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  • 6,334
9 votes
Accepted

Portfolio construction in reality?

Portfolio optimalisation depends heavily on the estimation of the moments (and therefore has HUGE estimation uncertainty). Even though it's useful for comparing and analysing different existing ...
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  • 333
8 votes

Calculate correlation between two sub portfolios and the combined portfolio

To clarify notation, you have an universe of $n=2000 \space$ stocks and two portfolio vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}$ with $\left\|\mathbf{a}\right\|_{1}=\left\|\mathbf{b}\right\|_{1}...
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8 votes
Accepted

Why do anomalies disappear after they get detected?

The best explanation I have seen so far is the so-called Adaptive Market Hypothesis by Andrew Lo: The adaptive market hypothesis, as proposed by Andrew Lo, is an attempt to reconcile economic ...
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  • 26.9k
8 votes
Accepted

How are modern portfolio theory (MPT) and CAPM related?

CAPM states that the expected return of any given asset should equal $ER_i=R_f+β_i (R_m-R_f)$, with α being the error term of the previous equation. Now, as α has an expected value of zero, then only ...
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  • 496
7 votes
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Optimal Portfolios

As a practitioner, I have worked on the following Maximize Yield/OAS for a Fixed Income Portfolio keeping the Rates Duration (Key Rate Durations) and Spread duration in a constrained range . There ...
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  • 883
7 votes
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On learning the bayesian approach to portfolio optimization

An introductory presentation by Michael Brandt from a seminar of Inquire Europe is Bayesian Portfolio Construction. His review Portfolio Choice Problems has a section on decision theory which could ...
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  • 866
7 votes
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Why is the variance of a portfolio a quadratic form?

if you take the variance of a single asset it scales as a quadratic, $$ var(\lambda X) = \lambda^2 var(X) $$ so it's not surprising that the general case gives a quadratic form.
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  • 6,743
7 votes

What’s the derivative of the sharpe ratio for one asset? Trying to optimize on it for a model

I agree that the paper could be much clearer: what it calls the “Sharp ratio derivative” is actually the “differential Sharpe ratio” proposed in a NIPS paper by Moody & Safell. In Section 2.2 of ...
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  • 200
7 votes

Most significant research articles for practical investors with research perspectives

A lot has happened since Markowitz and Sharpe. While their work is still considered foundational, the empirical/practical relevance of their models has been questioned by later work. Here are a few ...
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  • 9,077
7 votes

Portfolio Optimization and Global Minimum Variance Portfolio (GMV)

1) To be honest, any horizon is problematic in this respect. Simple sampling statistics 101 will tell you that the standard error around any estimate of true mean returns is the root time * variance. ...
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  • 4,896
7 votes

Closed-form analytical solution for the variance of the minimum-variance portfolio?

A few more steps beyond your last equation gives the answer. With $C = \mathbf{1}^T\mathbf{\Sigma}^{-1}\mathbf{1}$, we have $$\sigma_P^2 = [C^{-1} \mathbf{\Sigma}^{-1}\mathbf{1}]^T \mathbf{\Sigma} [C^{...
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  • 3,250
6 votes
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Difference between Sharpe Ratio and Information Ratio

Sharpe's 1966 equation had $R_b$ defined as the risk free rate. Looks like that was revised in 1994 to the 'reference benchmark', making the formulas essentially equivalent. If we refer to the ...
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  • 344
6 votes
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Mean Variance Portfolio theory and real-world problem?

Mean-variance (MV) is a framework rather than a prescription. This framework allows one to make, discuss, and defend his investment decision. In practice, there are many ways to make adjustments to ...
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6 votes
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Maximum Certainty Equivalent Portfolio with Transaction Costs

Seems like a small mistake in the last equation. It should read $\Delta^* = A^{-1} \left[\mu-\gamma \Sigma \omega_c - \frac{1}{\iota'A^{-1}\iota} \iota' A^{-1}(\mu-\gamma \Sigma \omega_c )\iota\...
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  • 116
6 votes
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Marginal Risk Contribution Formula

concerning your first question: the derivative does not disappear: $\sigma(R_p)$ contains the square root. To be more precise, set $$ \sigma(R_p) = \sqrt{w_1^2\cdot\sigma(R_1)^2 + w_2^2\cdot\sigma(R_2)...
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  • 1,386
6 votes
Accepted

Rockafellar-Uryasev mean-CVaR optimiztion

$VaR_\alpha$ is a scalar choice variable in the minimization problem. In the Rockafeller-Uryasev paper, it is simply called $\alpha\in R$. (C.f., the program described in Theorem 2 of that paper, or ...
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  • 393
6 votes
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Portfolio Optimization sum of weights constraint with short selling

In the early days of Portfolio Theory there were different views about short positions. Some authors modeled short positions as negative and required all weights to add up to 1 (first equation), ...
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  • 9,462
6 votes

Contribution of an asset's variance to portfolio variance

In this answer, I am assuming that you want to keep correlations constant. To begin with, note that the $N\times N$ covariance matrix $\Sigma$ with element $\Sigma_{i,j}=Cov(x_i,x_j)$ can be written ...
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  • 5,763
5 votes
Accepted

reference question about portfolio optimization

There are plenty of books on portfolio issues built according to formula "some theory + some R code (or Matlab, or S - which is very similar to R)". See for example Pfaff B. Financial Risk Modelling ...
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5 votes

Does Modern Portfolio Theory align with EMH?

The weak EMH states that it is impossible to earn an excess return given publicly known information such as past prices. Clearly, different securities have different expected returns. For example: the ...
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  • 7,652
5 votes

Create optimal portfolio by Treynor and Jensens Alpha

This optimization is trivial $$ w^{T,J}_i = \begin{cases} 1 \quad \text{if } i=\arg \max_i R^{T,J}(S_i) \\0 \quad \text{otherwise} \end{cases} $$ That is to say, when you optimize only one weight ...
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  • 14.4k
5 votes

Which algorithms do robo-advisors use?

Well, I did some modest research on this topic, looking at peers. Most of them use Modern Portfolio Theory, see this pic: You can find this small survey here: https://www.linkedin.com/pulse/...
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  • 51
5 votes

Basic question on Portfolio Theory

Of course estimating expected returns is the very core of portfolio management. Finding a useful covariance matrix too. To find both fills a book. So I first thought about closing the question. But it ...
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  • 13.3k
5 votes
Accepted

Interpretation of portfolio standard deviation

It depends on the distribution of the returns. If you assume that it's roughly normally distributed, then you have a ~68% chance for a return in the range of 1 standard deviation, ~95% chance for 2 ...
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  • 758
5 votes
Accepted

how can we know the residual return will be uncorrelated with the market return

Let us ignore the riskless rate for simplicity of the presentation. If you have (historical or simulated) return series $r_i$ for the portfolio and $r_i^M$ for the market, then the beta is the OLS ...
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  • 13.3k
5 votes
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Optimisation with strong correlated Assets

This answer will try and outline all the different possibilities I came across over the last couple of years, including drawbacks. But first, let me outline the problem a little. To appreciate the ...
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  • 2,874
5 votes

Why do anomalies disappear after they get detected?

Any anomaly that can be phrased as a "mispricing" or "relative value" opportunity can be expected to disappear as more people discover it and trade on it. For example, say that stock movements over ...
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  • 5,628
5 votes
Accepted

Finding a minimum variance portfolio when using a regulariser?

You're not going to get an analytic formula except in special cases of function $\rho(x)$. And you're probably going to want $\rho$ convex. If $\rho$ is convex, the problem is a convex optimization ...
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